Function theory of one complex variable:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2002
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate studies in mathematics
40 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xix, 502 p. ill. : 26 cm |
| ISBN: | 082182905X |
Internformat
MARC
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| 245 | 1 | 0 | |a Function theory of one complex variable |c Robert E. Greene and Steven G. Krantz |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2002 | |
| 300 | |a xix, 502 p. |b ill. : 26 cm | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 40 | |
| 500 | |a Includes bibliographical references and index | ||
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| 650 | 7 | |a FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS |2 larpcal | |
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Datensatz im Suchindex
| _version_ | 1804129323417862144 |
|---|---|
| adam_text | Contents
Preface to the Second Edition xv
Preface to the First Edition xvii
Acknowledgments xix
Chapter 1. Fundamental Concepts 1
§1.1. Elementary Properties of the Complex Numbers 1
§1.2. Further Properties of the Complex Numbers 3
§1.3. Complex Polynomials 10
§1.4. Holomorphic Functions, the Cauchy Riemann
Equations, and Harmonic Functions 14
§1.5. Real and Holomorphic Antiderivatives 17
Exercises 20
Chapter 2. Complex Line Integrals 29
§2.1. Real and Complex Line Integrals 29
§2.2. Complex Differentiability and Conformality 34
§2.3. Antiderivatives Revisited 40
§2.4. The Cauchy Integral Formula and the Cauchy
Integral Theorem 43
§2.5. The Cauchy Integral Formula: Some Examples 50
ix
x Contents
§2.6. An Introduction to the Cauchy Integral Theorem
and the Cauchy Integral Formula for More
General Curves 53
Exercises 60
Chapter 3. Applications of the Cauchy Integral 69
§3.1. Differentiability Properties of Holomorphic Functions 69
§3.2. Complex Power Series 74
§3.3. The Power Series Expansion for a Holomorphic Function 81
§3.4. The Cauchy Estimates and Liouville s Theorem 85
§3.5. Uniform Limits of Holomorphic Functions 88
§3.6. The Zeros of a Holomorphic Function 90
Exercises 94
Chapter 4. Meromorphic Functions and Residues 105
§4.1. The Behavior of a Holomorphic Function Near
an Isolated Singularity 105
§4.2. Expansion Around Singular Points 109
§4.3. Existence of Laurent Expansions 113
§4.4. Examples of Laurent Expansions 119
§4.5. The Calculus of Residues 122
§4.6. Applications of the Calculus of Residues to the
Calculation of Definite Integrals and Sums 128
§4.7. Meromorphic Functions and Singularities at Infinity 137
Exercises 145
Chapter 5. The Zeros of a Holomorphic Function 157
§5.1. Counting Zeros and Poles 157
§5.2. The Local Geometry of Holomorphic Functions 162
§5.3. Further Results on the Zeros of Holomorphic Functions 166
§5.4. The Maximum Modulus Principle 169
§5.5. The Schwarz Lemma 171
Exercises 174
Contents xi
Chapter 6. Holomorphic Functions as Geometric Mappings 179
§6.1. Biholomorphic Mappings of the Complex Plane
to Itself 180
§6.2. Biholomorphic Mappings of the Unit Disc to Itself 182
§6.3. Linear Fractional Transformations 184
§6.4. The Riemann Mapping Theorem: Statement and
Idea of Proof 189
§6.5. Normal Families 192
§6.6. Holomorphically Simply Connected Domains 196
§6.7. The Proof of the Analytic Form of the Riemann
Mapping Theorem 198
Exercises 202
Chapter 7. Harmonic Functions 207
§7.1. Basic Properties of Harmonic Functions 208
§7.2. The Maximum Principle and the Mean Value Property 210
§7.3. The Poisson Integral Formula 212
§7.4. Regularity of Harmonic Functions 218
§7.5. The Schwarz Reflection Principle 220
§7.6. Harnack s Principle 224
§7.7. The Dirichlet Problem and Subharmonic Functions 227
§7.8. The Perron Method and the Solution of the
Dirichlet Problem 236
§7.9. Conformal Mappings of Annuli 240
Exercises 243
Chapter 8. Infinite Series and Products 255
§8.1. Basic Concepts Concerning Infinite Sums and Products 255
§8.2. The Weierstrass Factorization Theorem 263
§8.3. The Theorems of Weierstrass and Mittag Leffler:
Interpolation Problems 266
Exercises 275
Chapter 9. Applications of Infinite Sums and Products 279
xii Contents
§9.1. Jensen s Formula and an Introduction to Blaschke
Products 279
§9.2. The Hadamard Gap Theorem 285
§9.3. Entire Functions of Finite Order 288
Exercises 296
Chapter 10. Analytic Continuation 299
§10.1. Definition of an Analytic Function Element 299
§10.2. Analytic Continuation Along a Curve 305
§10.3. The Monodromy Theorem 307
§10.4. The Idea of a Riemann Surface 310
§10.5. The Elliptic Modular Function and Picard s Theorem 314
§10.6. Elliptic Functions 323
Exercises 330
Chapter 11. Topology 335
§11.1. Multiply Connected Domains 335
§11.2. The Cauchy Integral Formula for Multiply
Connected Domains 338
§11.3. Holomorphic Simple Connectivity and Topological
Simple Connectivity 343
§11.4. Simple Connectivity and Connectedness of the
Complement 344
§11.5. Multiply Connected Domains Revisited 349
Exercises 352
Chapter 12. Rational Approximation Theory 361
§12.1. Runge s Theorem 361
§12.2. Mergelyan s Theorem 367
§12.3. Some Remarks about Analytic Capacity 376
Exercises 379
Chapter 13. Special Classes of Holomorphic Functions 383
§13.1. Schlicht Functions and the Bieberbach Conjecture 384
Contents xiii
§13.2. Continuity to the Boundary of Conformal Mappings 390
§13.3. Hardy Spaces 399
§13.4. Boundary Behavior of Functions in Hardy Classes
[An Optional Section for Those Who Know
Elementary Measure Theory] 404
Exercises 410
Chapter 14. Hilbert Spaces of Holomorphic Functions,
the Bergman Kernel, and Biholomorphic
Mappings 413
§14.1. The Geometry of Hilbert Space 413
§14.2. Orthonormal Systems in Hilbert Space 424
§14.3. The Bergman Kernel 429
§14.4. Bell s Condition R 435
§14.5. Smoothness to the Boundary of Conformal Mappings 441
Exercises 444
Chapter 15. Special Functions 447
§15.1. The Gamma and Beta Functions 447
§15.2. The Riemann Zeta Function 455
Exercises 465
Chapter 16. The Prime Number Theorem 469
§16.0. Introduction 469
§16.1. Complex Analysis and the Prime Number Theorem 471
§16.2. Precise Connections to Complex Analysis 476
§16.3. Proof of the Integral Theorem 481
Exercises 482
APPENDIX A: Real Analysis 485
APPENDIX B: The Statement and Proof of Goursafs Theorem 491
References 495
Index 499
|
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| author | Greene, Robert Everist 1943- Krantz, Steven G. 1951- |
| author_GND | (DE-588)130595063 (DE-588)130535907 |
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| discipline | Mathematik |
| edition | 2. ed. |
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| id | DE-604.BV014534377 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:03:19Z |
| institution | BVB |
| isbn | 082182905X |
| language | English |
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| physical | xix, 502 p. ill. : 26 cm |
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| publisher | American Mathematical Society |
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| series | Graduate studies in mathematics |
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| spelling | Greene, Robert Everist 1943- Verfasser (DE-588)130595063 aut Function theory of one complex variable Robert E. Greene and Steven G. Krantz 2. ed. Providence, RI American Mathematical Society 2002 xix, 502 p. ill. : 26 cm txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 40 Includes bibliographical references and index ANÁLISE MATEMÁTICA larpcal FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS larpcal Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Komplexe Variable (DE-588)4164905-9 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Komplexe Variable (DE-588)4164905-9 s Funktion Mathematik (DE-588)4071510-3 s 1\p DE-604 Funktionentheorie (DE-588)4018935-1 s 2\p DE-604 Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Graduate studies in mathematics 40 (DE-604)BV009739289 40 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009890284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Greene, Robert Everist 1943- Krantz, Steven G. 1951- Function theory of one complex variable Graduate studies in mathematics ANÁLISE MATEMÁTICA larpcal FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS larpcal Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik (DE-588)4071510-3 gnd Komplexe Variable (DE-588)4164905-9 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4071510-3 (DE-588)4164905-9 (DE-588)4018935-1 |
| title | Function theory of one complex variable |
| title_auth | Function theory of one complex variable |
| title_exact_search | Function theory of one complex variable |
| title_full | Function theory of one complex variable Robert E. Greene and Steven G. Krantz |
| title_fullStr | Function theory of one complex variable Robert E. Greene and Steven G. Krantz |
| title_full_unstemmed | Function theory of one complex variable Robert E. Greene and Steven G. Krantz |
| title_short | Function theory of one complex variable |
| title_sort | function theory of one complex variable |
| topic | ANÁLISE MATEMÁTICA larpcal FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS larpcal Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik (DE-588)4071510-3 gnd Komplexe Variable (DE-588)4164905-9 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | ANÁLISE MATEMÁTICA FUNÇÕES DE VÁRIAS VARIÁVEIS COMPLEXAS Fonctions d'une variable complexe Functions of complex variables Funktion Mathematik Komplexe Variable Funktionentheorie |
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